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The Mathematical Explorer

Stan Wagon and Wolfram Research
Wolfram Research
Publication Date: 
Student Helps
[Reviewed by
Marvin Schaefer
, on

Readers of Read This! are probably familiar with Stan Wagon as a prolific author and educator. Many of his papers have appeared in the Monthly and Mathematics Magazine, and the MAA has published two of his seven books: Which Way Did the Bicycle Go? (with J. D. E. Konhauser and Dan Velleman) and Old and New Unsolved Problems in Plane Geometry and Number Theory (with Victor Klee).

The Mathematical Explorer is an electronic book divided into 15 chapters: Prime Numbers, Calculus, Formulas for Computing Pi, Square Wheels, The Power of Check Digits, Secret Codes, Recreational Mathematics, Exploring Escher Patterns, Varieties of Roses, Turtle Fractalization, Patterns in Chaos, Fermat's Last Theorem, Riemann Hypothesis, Unusual Number Systems, and The Four Color Theorem. Each chapter has several subchapters. Far more mathematical topics are covered than is suggested by the chapter titles, including: continued fractions, Diophantine equations, modular arithmetic, the Buffon Needle Problem, Fibonacci numbers, the Brachistochrone Problem, etc.

The purpose of The Mathematical Explorer is explained in its Introduction:

The Mathematical Explorer is an interactive journey through some of the most fascinating problems in the history of mathematics — problems that have challenged mathematicians from the ancient Greeks up to the modern day. It includes topics on questions that were only very recently solved, such as Fermat's Last Theorem and the computer proof of the Four-Color Theorem, and also explores as yet unsolved problems such as the Riemann Hypothesis...

The treatment of each topic is designed to be educational as well as entertaining; it includes a clear explanation of the important concepts along with fascinating cultural and historical details. Many topics have a strong computational thread, while still others are best understood through graphical visualization. Integrated with The Mathematical Explorer are a powerful computational engine and interface that rely on technology from the creators of Mathematica ... With The Mathematical Explorer, you can perform a wide range of numerical and symbolic calculations as well as create an unlimited array of graphics to help you better understand the concepts you are exploring.

The Mathematical Explorer is intended as an open-ended, interactive resource to the world of modern mathematics, one that allows you to walk in the computational footsteps of the great mathematicians and experience the wonder of discovery that has fascinated amateurs and professionals alike throughout the ages.

The degree to which Mathematical Explorer meets these expressed goals varies according to the selected topic. The packaging does not identify the minimal level of mathematical sophistication expected of the user — it is certainly accessible to an undergraduate or advanced high school student who has learnt the mechanics of elementary calculus and a few trigonometric identities. The text is contained in a series of Mathematica notebooks that are supported by a restricted configuration of the Mathematica kernel. The Mathematical Explorer is the first "Mathematica-powered" product Wolfram produced. It is made available at a very small fraction of the commercial price for Mathematica (and is much less expensive than student versions of Mathematica). As has been demonstrated on Ed Pegg's recreational mathematics website (, Mathematical Explorer can be used to create a very impressive variety of new and useful notebooks that make use of the computational, graphics and manipulative functions. So, in a sense, the purchaser of Mathematical Explorer gets a powerful "mini" version of Mathematica as a side-benefit that will enable many independent forays into a variety of topics.

The interactive portions of Mathematical Explorer are expressed using Mathematica's "standard form" notation, a notation that often differs from conventional mathematical notation only in capitalization and the use of square brackets for parentheses in functional notation. All symbolic computational results are formatted into the more familiar "traditional" notation. While Mathematical Explorer's restricted kernel does not support all of the functionality of Mathematica, it is quite sufficient to allow the user to perform a very wide range of experiments and computations.

In order to explore or experiment, the user is directed to click on a Mathematica expression (or program) which is activated by holding down the Shift key and pressing Enter. The results can be amazing — graphs and tables are created, often a complicated algebraic expression is manipulated and simplified, eliminating tedious manipulations. But sometimes the result of doing this appears to be pointless, as the result is already printed below the expression (e.g., FactorInteger[265252859812191058636308480000000]//FactorForm instantly reproduces the already displayed 226 314 57 74 112 132 17 19 23 29). The inquisitive user will soon get the general idea and may take the initiative to insert his own parameters perform further experiments of his own crafting. The user may sometimes conclude that there is no need to evaluate such expressions — modern computers are so fast that the result appears immediately and nothing of interest happens. However, it is sometimes essential that the user perform this ritual in order to enable subsequent computations that depend on its results.

The text is written in a light, friendly tone, and the reader is challenged with a sprinkling of exercises. The reader is guided swiftly through a brief historical context toward quick derivations and computational results. Now and then, a quick computation in Mathematical Explorer is used to "prove" some fact (like primality) or to assert the finite sum of an infinite series. But the text also exhibits counterexamples in order to stress the fact that computational evidence, by itself, does not provide proof.

Wagon presents a number of beautiful intuitive explanations, insights and motivating derivations. For example, he shows an intuitive overview derivation of why the Leibniz series sums to π/4 by starting with a geometric series that yields 1/(1+x2) on which he then performs term-by-term integration of both sides to find a series expansion for tan-1(x). He uses Mathematica at several steps of the derivation, exhibits partial sums as motivating evidence, and states clearly that he is leaving out some important proof details in order to get the point across. This seems to be appropriate for introducing these topics to the young reader.

Mathematical Explorer "struts its stuff" beautifully in its explorations of topics from number theory and combinatorics. There are many unexpected gems in the text (e.g., Peano's space-filling curve turns up unexpectedly in the chapter on Turtle Fractalization, the dihedral group is shown to provide the basis for check digits on German currency!). I particularly enjoyed the chapters on the Four Color Theorem and on exploring the group theory behind Escher's 1938 "basket-weave" tilings. In both of these chapters, history and experimentation are stressed. Wagon introduces the reader to planar graph theory and then presents a motivated series of overviews of failed proof attempts. The user is given sample maps on which to try finding a 4-coloring, and is invited to experiment with coloring algorithms. This chapter makes very effective use of Mathematica's animation capabilities — although I had to use Mathematical Explorer's controls to slow down the action which ran too rapidly on my 1998 computer. Wagon's discussion of Escher tilings ends up focusing on the combinatorial problem of choosing colors for strands in the basic pattern so that the strands appear to be a continuous weaving in instantiations of repeated (m × n)-unit tilings.

The chapter, on Fermat's Last Theorem, provides delightful tours of linear and quadratic Diophantine Equations (and puzzles!), of Pythagorean triples, and Pell's equation. It gives a good overview of much of the mathematics that has come from investigation of these topics, but does not try to go into modular forms and other mathematics needed in Wiles' proof. The chapter on the Riemann hypothesis gives a notably clear explanation of the hypothesis, explores the real, negative, and full Zeta function with graphic explorations of zeroes, and presents splendid examples of the Riemann hypothesis' importance in prime number theory. The chapter on Square Wheels investigates cycloids and catenaries, demonstrating their properties through animations. It includes the [in]famous photograph of a smiling Stan Wagon having a smooth ride on his square-wheeled bicycle over a track of inverted catenaries.

Some of the chapters appear to have been too-quickly written. I was disappointed with the calculus chapter's treatment of derivatives and integrals. Although the chapter served well to demonstrate Mathematical Explorer's power to perform differentiation, antidifferentiation, solving a differential equation and evaluating definite integrals, these subsections did not appear to inspire one to want to explore and learn new concepts as the other chapters did.

The speed of modern computers may undermine some of the value of the explorations. I've been impressed with the power of symbolic tools from the time I first used an early version of Mathematica. I just asked Mathematical Explorer to display the 12307 places of π I requested at random, and it did, using 0.16 seconds of computation on my elderly computer to do so! Solutions to systems of linear Diophantine equations, to graph coloring problems, and to finding continued fraction expansions come about very quickly — so quickly that it is not clear to me that use of Mathematical Explorer alone will develop the intuitive experiences that should accompany mathematical discovery.

Several very powerful tools are provided to the user in the form of functions (e.g., LinearDiophantineSolve will provide general solutions to linear Diophantine equations in two variables). However, the principles behind the function are not explained, and no particular guidance is provided for the interested user. Users already experienced in Mathematica will know that they can sometimes see the code. (I.e., this can be done by typing ??LinearDiophantineSolve), but the code is somewhat difficult to read because of the universal use of private contextual pathnames prepended onto variable and function names. Wagon's code occasionally uses rather sophisticated coding conventions designed for computational efficiency. To those users, the obvious next step is obtaining a copy of one of Wagon's Mathematica-oriented books such as Mathematica in Action. Note that Mathematical Explorer's restricted kernel, most likely will not run all the programs contained in such books. And, in my opinion, people who already have Mathematica who would like to see Wagon's full derivations, code, and many of the proofs, would be best advised to purchase Mathematica in Action in the first place.

Mathematical Explorer comes on a CD that can be used on either a Macintosh or Windows computer. There is a small Getting Started pamphlet that explains installation and concisely provides a great deal of essential usage information — far more than one might expect from its sparsely-printed 36-page format. As a Mathematica user, I looked at the installation instructions and straight-away began working my way through Mathematical Explorer's chapters. But I soon began encountering expressions written in Mathematica's syntax, including several that made use of options or Mathematica's pure function notations. I wondered how well a reader who had never experienced Mathematica before would fare with this product, and finally consulted this helpful booklet. I believe that it contains adequate information to guide the novice past the initial obstacles. It also provides advice on using the Help Browser's Master Index to find explanations on many of the features that Wagon uses in his exposition. The contents of Getting Started are also accessible on-line through the Help Browser. (The package also includes Wolfram's restrictive license agreement and product registration forms.)

Since this is an electronic textbook, the user is encouraged to experiment freely with the formulas and plotting expressions provided for the purpose. Wagon frequently provides a variety of powerful functions, complete with usage information, designed for experimentation. The user may also create and evaluate new expressions of his own crafting; he may also annotate the text (however, see below). This very powerful capability should serve as a strong stimulus for gaining insight into the underlying structures the text touches on.

Just as in Mathematica, the Mathematical Explorer allows the user to create, open, save and print notebooks containing text intermixed with invocations of a useful subset of Mathematica's rich functionality. The intended primary user interface to Mathematical Explorer's electronic text is through Mathematica's Help Browser. This allows one to select chapters, their sections and subsections through a mouse-click interface. It also allows one to search for concepts, mathematician biographies, references, etc., and it also provides reference material on much of Mathematica itself. The term 'browser' suggests something akin to familiar Internet interfaces like Netscape or Internet Explorer. But it is more primitive, and can neither spawn identical copies of itself as separate windows, provide for creating bookmarks, etc. It offers a Back button, but there is no corresponding Forward button. Also, the Back button does not preserve and restore context. As a consequence, it can be somewhat awkward to switch repeatedly between two or more related sections of the text.

The text frequently contains links to references, solutions to exercises, biographies of mathematicians, etc. Useful as these features and cross-references are, whenever a user follows a link he also loses the electronic context he may have established on the page he leaves for the purpose. Indeed, returning to that page with the Back button returns one to the beginning of the page, and all computations or annotations made on that page are gone! Getting Started suggests that a user select and copy text into a new notebook if he intends to make lasting experiments or annotations. The new notebook can be saved and restored for use over a series of sessions. However, some of Mathematical Explorer's presentation formatting and the electronic links will be lost in the process of copying text using the method explained in the booklet.

Also, some of the exemplary Mathematica code will prove to be opaque to users not already familiar with Mathematica's coding conventions. For example, expressions such as SelectRange[100],Length[SumOfSquaresRepresentations[2,#]]>1&] or Select[Range[500], !(IntegerQ[#])&&Length[SumOfSquaresRepresentations[2,#]]]>1&] use Mathematica's pure function notation, which is not discussed in the examples. The term function and the notations # and & can be found, of course, in the on-line Reference Guide (but not in the Index), but the main body of the explanation of pure functions (another term that cannot be found in the Help Index) requires access either to the Mathematica Book or to the version at the Wolfram Website, a potential problem to users who cannot easily go online.

Much of Mathematical Explorer's content has been profitably drawn from Wagon's earlier writings. These sources include Mathematica in Action, Which Way did the Bicycle Go, and A Course in Computational Number Theory (with David Bressoud). Mathematical Explorer comes with a splendid set of printed and Internet references for further study.

As one has come to expect, Prof. Wagon is a splendid writer and educator. There is a tremendous variety of good mathematics in the Mathematical Explorer, and users are certain to find exciting mathematical concepts, insights and challenges aplenty.

Marvin Schaefer ( is a computer security expert and was chief scientist at the National Computer Security Center at the NSA, and at Arca Systems. He has been a member of the MAA for 39 years and now operates an antiquarian book store called Books With a Past.

The table of contents is not available.